It is an essential concept in computer science and is widely used in various algorithms, including searching, sorting, and traversing data structures. See your article appearing on the GeeksforGeeks main page. However -these are constant number of ops, while not changing the number of "iterations". It is not the very best in terms of performance but more efficient traditionally than most other simple O (n^2) algorithms such as selection sort or bubble sort. Credit : Stephen Halim. A function that calls itself directly or indirectly is called a recursive function and such kind of function calls are called recursive calls. The Space Complexity is O(N) and the Time complexity is O(2^N) because the root node has 2 children and 4 grandchildren. Recursion is slower than iteration since it has the overhead of maintaining and updating the stack. Time Complexity: Very high. – Bernhard Barker. The auxiliary space required by the program is O(1) for iterative implementation and O(log 2 n) for. Since you cannot iterate a tree without using a recursive process both of your examples are recursive processes. First, one must observe that this function finds the smallest element in mylist between first and last. The two features of a recursive function to identify are: The tree depth (how many total return statements will be executed until the base case) The tree breadth (how many total recursive function calls will be made) Our recurrence relation for this case is T (n) = 2T (n-1). Similarly, Space complexity of an algorithm quantifies the amount of space or memory taken by an algorithm to run as a function of the length of the input. Next, we check to see if number is found in array [index] in line 4. Improve this question. 1. e. e. In terms of time complexity and memory constraints, iteration is preferred over recursion. Iteration. Space The Fibonacci sequence is de ned: Fib n = 8 >< >: 1 n == 0Efficiency---The Time Complexity of an Algorithm In the bubble sort algorithm, there are two kinds of tasks. To understand what Big O notation is, we can take a look at a typical example, O (n²), which is usually pronounced “Big O squared”. That means leaving the current invocation on the stack, and calling a new one. Memoization is a method used to solve dynamic programming (DP) problems recursively in an efficient manner. The reason why recursion is faster than iteration is that if you use an STL container as a stack, it would be allocated in heap space. Recursion can be replaced using iteration with stack, and iteration can also be replaced with recursion. Iteration: An Empirical Study of Comprehension Revisited. The only reason I chose to implement the iterative DFS is that I thought it may be faster than the recursive. "Recursive is slower then iterative" - the rational behind this statement is because of the overhead of the recursive stack (saving and restoring the environment between calls). GHCRecursion is the process of calling a function itself repeatedly until a particular condition is met. Both are actually extremely low level, and you should prefer to express your computation as a special case of some generic algorithm. Auxiliary Space: DP may have higher space complexity due to the need to store results in a table. What will be the run time complexity for the recursive code of the largest number. 5: We mostly prefer recursion when there is no concern about time complexity and the size of code is. In this Video, we are going to learn about Time and Space Complexities of Recursive Algo. In the factorial example above, we have reached the end of our necessary recursive calls when we get to the number 0. Can be more complex and harder to understand, especially for beginners. The reason is because in the latter, for each item, a CALL to the function st_push is needed and then another to st_pop. The basic idea of recursion analysis is: Calculate the total number of operations performed by recursion at each recursive call and do the sum to get the overall time complexity. There are many other ways to reduce gaps which leads to better time complexity. Any function that is computable – and many are not – can be computed in an infinite number. Recursion is quite slower than iteration. But, if recursion is written in a language which optimises the. Iteration and recursion are two essential approaches in Algorithm Design and Computer Programming. Moreover, the recursive function is of exponential time complexity, whereas the iterative one is linear. e. How many nodes are there. High time complexity. Iterative Backtracking vs Recursive Backtracking; Time and Space Complexity; Introduction to Iteration. If time complexity is the point of focus, and number of recursive calls would be large, it is better to use iteration. Whenever you are looking for time taken to complete a particular algorithm, it's best you always go for time complexity. So let us discuss briefly on time complexity and the behavior of Recursive v/s Iterative functions. Recursion vs. Iteration reduces the processor’s operating time. The recursive function runs much faster than the iterative one. The difference between O(n) and O(2 n) is gigantic, which makes the second method way slower. The actual complexity depends on what actions are done per level and whether pruning is possible. Both recursion and ‘while’ loops in iteration may result in the dangerous infinite calls situation. That's a trick we've seen before. Singly linked list iteration complexity. A recursive function is one that calls itself, such as the printList function which uses the divide and conquer principle to print the numbers 1 to 5. For example, using a dict in Python (which has (amortized) O (1) insert/update/delete times), using memoization will have the same order ( O (n)) for calculating a factorial as the basic iterative solution. Both involve executing instructions repeatedly until the task is finished. It allows for the processing of some action zero to many times. mov loopcounter,i dowork:/do work dec loopcounter jmp_if_not_zero dowork. I tried check memory complexity for recursive and iteration program computing factorial. Its time complexity anal-ysis is similar to that of num pow iter. Iteration and recursion are key Computer Science techniques used in creating algorithms and developing software. Iteration: Generally, it has lower time complexity. Below is the implementation using a tail-recursive function. Determine the number of operations performed in each iteration of the loop. To my understanding, the recursive and iterative version differ only in the usage of the stack. No. For. There are O(N) recursive calls in our recursive approach, and each call uses O(1) operations. Recursion adds clarity and reduces the time needed to write and debug code. e. But recursion on the other hand, in some situations, offers convenient tool than iterations. The reason for this is that the slowest. It breaks down problems into sub-problems which it further fragments into even more sub. Also remember that every recursive method must make progress towards its base case (rule #2). They can cause increased memory usage, since all the data for the previous recursive calls stays on the stack - and also note that stack space is extremely limited compared to heap space. but this is a only a rough upper bound. Infinite Loop. The previous example of O(1) space complexity runs in O(n) time complexity. The recursive function runs much faster than the iterative one. As you correctly noted the time complexity is O (2^n) but let's look. Our iterative technique has an O(N) time complexity due to the loop's O(N) iterations (N). There is an edge case, called tail recursion. Recursion • Rules" for Writing Recursive Functions • Lots of Examples!. Time and Space Optimization: Recursive functions can often lead to more efficient algorithms in terms of time and space complexity. An iteration happens inside one level of. , a path graph if we start at one end. Naive sorts like Bubble Sort and Insertion Sort are inefficient and hence we use more efficient algorithms such as Quicksort and Merge Sort. Recurson vs Non-Recursion. Major difference in time/space complexity between code running recursion vs iteration is caused by this : as recursion runs it will create new stack frame for each recursive invocation. However, the iterative solution will not produce correct permutations for any number apart from 3 . In order to build a correct benchmark you must - either chose a case where recursive and iterative versions have the same time complexity (say linear). Reduced problem complexity Recursion solves complex problems by. Time complexity is very high. There are often times that recursion is cleaner, easier to understand/read, and just downright better. Add a comment. If it is, the we are successful and return the index. Let's try to find the time. When recursion reaches its end all those frames will start unwinding. The Java library represents the file system using java. Sometimes it's even simpler and you get along with the same time complexity and O(1) space use instead of, say, O(n) or O(log n) space use. Generally, it has lower time complexity. def function(): x = 10 function() When function () executes the first time, Python creates a namespace and assigns x the value 10 in that namespace. To visualize the execution of a recursive function, it is. You can find a more complete explanation about the time complexity of the recursive Fibonacci. If you're wondering about computational complexity, see here. Auxiliary Space: O(N), for recursion call stack If you like GeeksforGeeks and would like to contribute, you can also write an article using write. g. 10 Answers Sorted by: 165 Recursion is usually much slower because all function calls must be stored in a stack to allow the return back to the caller functions. Recursion terminates when the base case is met. when recursion exceeds a particular limit we use shell sort. Data becomes smaller each time it is called. Recursion takes longer and is less effective than iteration. While the results of that benchmark look quite convincing, tail-recursion isn't always faster than body recursion. It is fast as compared to recursion. the last step of the function is a call to the. , referring in part to the function itself. Utilization of Stack. Insertion sort is a stable, in-place sorting algorithm that builds the final sorted array one item at a time. This involves a larger size of code, but the time complexity is generally lesser than it is for recursion. fib(n) grows large. Increment the end index if start has become greater than end. We still need to visit the N nodes and do constant work per node. Recursion vs Iteration: You can reduce time complexity of program with Recursion. Time Complexity. And I have found the run time complexity for the code is O(n). So it was seen that in case of loop the Space Complexity is O(1) so it was better to write code in loop instead of tail recursion in terms of Space Complexity which is more efficient than tail. I found an answer here but it was not clear enough. The total time complexity is then O(M(lgmax(m1))). And, as you can see, every node has 2 children. It consists of initialization, comparison, statement execution within the iteration, and updating the control variable. Improve this. 2. functions are defined by recursion, so implementing the exact definition by recursion yields a program that is correct "by defintion". Applying the Big O notation that we learn in the previous post , we only need the biggest order term, thus O (n). So when recursion is doing constant operation at each recursive call, we just count the total number of recursive calls. I would appreciate any tips or insights into understanding the time complexity of recursive functions like this one. Time Complexity of iterative code = O (n) Space Complexity of recursive code = O (n) (for recursion call stack) Space Complexity of iterative code = O (1). Performs better in solving problems based on tree structures. "Recursive is slower then iterative" - the rational behind this statement is because of the overhead of the recursive stack (saving and restoring the environment between calls). Iteration uses the CPU cycles again and again when an infinite loop occurs. Here are the general steps to analyze the complexity of a recurrence relation: Substitute the input size into the recurrence relation to obtain a sequence of terms. Calculate the cost at each level and count the total no of levels in the recursion tree. Iteration — Non-recursion. When considering algorithms, we mainly consider time complexity and space complexity. In our recursive technique, each call consumes O(1) operations, and there are O(N) recursive calls overall. Big O Notation of Time vs. A filesystem consists of named files. The recursive step is n > 0, where we compute the result with the help of a recursive call to obtain (n-1)!, then complete the computation by multiplying by n. A filesystem consists of named files. perf_counter() and end_time to see the time they took to complete. e. The iteration is when a loop repeatedly executes until the controlling condition becomes false. It can reduce the time complexity to: O(n. This is usually done by analyzing the loop control variables and the loop termination condition. Evaluate the time complexity on the paper in terms of O(something). In this case, iteration may be way more efficient. linear, while the second implementation is shorter but has exponential complexity O(fib(n)) = O(φ^n) (φ = (1+√5)/2) and thus is much slower. e. time complexity or readability but. Only memory for the. 11. remembering the return values of the function you have already. What this means is, the time taken to calculate fib (n) is equal to the sum of time taken to calculate fib (n-1) and fib (n-2). The recursive version’s idea is to process the current nodes, collect their children and then continue the recursion with the collected children. The towers of Hanoi problem is hard no matter what algorithm is used, because its complexity is exponential. Iterative Sorts vs. It consists of three poles and a number of disks of different sizes which can slide onto any pole. Yes, recursion can always substitute iteration, this has been discussed before. The primary difference between recursion and iteration is that recursion is a process, always. def tri(n: Int): Int = { var result = 0 for (count <- 0 to n) result = result + count result} Note that the runtime complexity of this algorithm is still O(n) because we will be required to iterate n times. In. ; Otherwise, we can represent pow(x, n) as x * pow(x, n - 1). In this tutorial, we’ll talk about two search algorithms: Depth-First Search and Iterative Deepening. A recursive algorithm can be time and space expensive because to count the value of F n we have to call our recursive function twice in every step. This is a simple algorithm, and good place to start in showing the simplicity and complexity of of recursion. Iterative Backtracking vs Recursive Backtracking; Time and Space Complexity; Introduction to Iteration. Hence, even though recursive version may be easy to implement, the iterative version is efficient. Sometimes the rewrite is quite simple and straight-forward. Both recursion and ‘while’ loops in iteration may result in the dangerous infinite calls situation. As a thumbrule: Recursion is easy to understand for humans. If a k-dimensional array is used, where each dimension is n, then the algorithm has a space. 1. I assume that solution is O(N), not interesting how implemented is multiplication. Since this is the first value of the list, it would be found in the first iteration. Introduction. Your stack can blow-up if you are using significantly large values. So it was seen that in case of loop the Space Complexity is O(1) so it was better to write code in loop instead of tail recursion in terms of Space Complexity which is more efficient than tail recursion. The idea is to use one more argument and accumulate the factorial value in the second argument. I'm a little confused. This reading examines recursion more closely by comparing and contrasting it with iteration. When you're k levels deep, you've got k lots of stack frame, so the space complexity ends up being proportional to the depth you have to search. Both approaches create repeated patterns of computation. In contrast, the iterative function runs in the same frame. There are factors ignored, like the overhead of function calls. Recursion: The time complexity of recursion can be found by finding the value of the nth recursive call in terms of the previous calls. Thus the amount of time. Here are the general steps to analyze the complexity of a recurrence relation: Substitute the input size into the recurrence relation to obtain a sequence of terms. So for practical purposes you should use iterative approach. Recursion involves creating and destroying stack frames, which has high costs. If a new operation or iteration is needed every time n increases by one, then the algorithm will run in O(n) time. Imagine a street of 20 book stores. org or mail your article to review-team@geeksforgeeks. Utilization of Stack. So for practical purposes you should use iterative approach. Iteration is generally faster, some compilers will actually convert certain recursion code into iteration. Iteration is always cheaper performance-wise than recursion (at least in general purpose languages such as Java, C++, Python etc. With this article at OpenGenus, you must have the complete idea of Tower Of Hanoi along with its implementation and Time and Space. Also, deque performs better than a set or a list in those kinds of cases. If the compiler / interpreter is smart enough (it usually is), it can unroll the recursive call into a loop for you. But it has lot of overhead. A method that requires an array of n elements has a linear space complexity of O (n). Line 6-8: 3 operations inside the for-loop. If the number of function. Iterative codes often have polynomial time complexity and are simpler to optimize. However, the space complexity is only O(1). Then we notice that: factorial(0) is only comparison (1 unit of time) factorial(n) is 1 comparison, 1 multiplication, 1 subtraction and time for factorial(n-1) factorial(n): if n is 0 return 1 return n * factorial(n-1) From the above analysis we can write:DFS. Sorted by: 1. So a filesystem is recursive: folders contain other folders which contain other folders, until finally at the bottom of the recursion are plain (non-folder) files. . So go for recursion only if you have some really tempting reasons. The major difference between the iterative and recursive version of Binary Search is that the recursive version has a space complexity of O(log N) while the iterative version has a space complexity of O(1). "tail recursion" and "accumulator based recursion" are not mutually exclusive. Iteration terminates when the condition in the loop fails. e execution of the same set of instructions again and again. First, you have to grasp the concept of a function calling itself. The major driving factor for choosing recursion over an iterative approach is the complexity (i. 2. Let’s take an example to explain the time complexity. Time Complexity: Time complexity of the above implementation of Shell sort is O(n 2). In the worst case scenario, we will only be left with one element on one far side of the array. Explanation: Since ‘mid’ is calculated for every iteration or recursion, we are diving the array into half and then try to solve the problem. It's less common in C but still very useful and powerful and needed for some problems. Use a substitution method to verify your answer". Its time complexity is easier to calculate by calculating the number of times the loop body gets executed. )) chooses the smallest of. It is faster because an iteration does not use the stack, Time complexity. And Iterative approach is always better than recursive approch in terms of performance. Frequently Asked Questions. Time Complexity: In the above code “Hello World” is printed only once on the screen. Consider for example insert into binary search tree. Usage: Recursion is generally used where there is no issue of time complexity, and code size requires being small. Recursion vs. Here N is the size of data structure (array) to be sorted and log N is the average number of comparisons needed to place a value at its right. Therefore, the time complexity of the binary search algorithm is O(log 2 n), which is very efficient. 12. We prefer iteration when we have to manage the time complexity and the code size is large. • Algorithm Analysis / Computational Complexity • Orders of Growth, Formal De nition of Big O Notation • Simple Recursion • Visualization of Recursion, • Iteration vs. Courses Practice What is Recursion? The process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. Time Complexity: O(n) Auxiliary Space: O(n) The above function can be written as a tail-recursive function. Iteration is preferred for loops, while recursion is used for functions. Time Complexity: O(N), to traverse the linked list of size N. The 1st one uses recursive calls to calculate the power(M, n), while the 2nd function uses iterative approach for power(M, n). What are the benefits of recursion? Recursion can reduce time complexity. 3. Recursion is inefficient not because of the implicit stack but because of the context switching overhead. I think that Prolog shows better than functional languages the effectiveness of recursion (it doesn't have iteration), and the practical limits we encounter when using it. Both iteration and recursion are. pop() if node. To visualize the execution of a recursive function, it is. The purpose of this guide is to provide an introduction to two fundamental concepts in computer science: Recursion and Backtracking. We can choose which to use either recursion or iteration, considering Time Complexity and size of the code. The puzzle starts with the disk in a neat stack in ascending order of size in one pole, the smallest at the top thus making a conical shape. For Example, the Worst Case Running Time T (n) of the MERGE SORT Procedures is described by the recurrence. O (n * n) = O (n^2). That’s why we sometimes need to. 5. Time complexity. Time complexity. Of course, some tasks (like recursively searching a directory) are better suited to recursion than others. Alternatively, you can start at the top with , working down to reach and . Program for Tower of Hanoi Algorithm; Time Complexity Analysis | Tower Of Hanoi (Recursion) Find the value of a number raised to its reverse; Recursively remove all adjacent duplicates; Print 1 to n without using loops; Print N to 1 without loop; Sort the Queue using Recursion; Reversing a queue using. Our iterative technique has an O(N) time complexity due to the loop's O(N) iterations (N). Thus, the time complexity of factorial using recursion is O(N). When a function is called, there is an overhead of allocating space for the function and all its data in the function stack in recursion. Share. Iteration is always faster than recursion if you know the amount of iterations to go through from the start. from collections import deque def preorder3(initial_node): queue = deque([initial_node]) while queue: node = queue. This is called a recursive step: we transform the task into a simpler action (multiplication by x) and a. It is. Hence, usage of recursion is advantageous in shorter code, but higher time complexity. 1Review: Iteration vs. The total number of function calls is therefore 2*fib (n)-1, so the time complexity is Θ (fib (N)) = Θ (phi^N), which is bounded by O (2^N). Recursion takes longer and is less effective than iteration. But then, these two sorts are recursive in nature, and recursion takes up much more stack memory than iteration (which is used in naive sorts) unless. The above code snippet is a function for binary search, which takes in an array, size of the array, and the element to be searched x. When it comes to finding the difference between recursion vs. difference is: recursive programs need more memory as each recursive call pushes state of the program into stack and stackoverflow may occur. Let’s write some code. What is the average case time complexity of binary search using recursion? a) O(nlogn) b) O(logn) c) O(n) d) O(n 2). What are the advantages of recursion over iteration? Recursion can reduce time complexity. (Think!) Recursion has a large amount of overhead as compared to Iteration. Backtracking always uses recursion to solve problems. We added an accumulator as an extra argument to make the factorial function be tail recursive. We often come across this question - Whether to use Recursion or Iteration. |. Recursion takes additional stack space — We know that recursion takes extra memory stack space for each recursive calls, thus potentially having larger space complexity vs. A single point of comparison has a bias towards the use-case of recursion and iteration, in this case; Iteration is much faster. Evaluate the time complexity on the paper in terms of O(something). I just use a normal start_time = time. : f(n) = n + f(n-1) •Find the complexity of the recurrence: –Expand it to a summation with no recursive term. It is faster than recursion. Share. I am studying Dynamic Programming using both iterative and recursive functions. When you have nested loops within your algorithm, meaning a loop in a loop, it is quadratic time complexity (O(n^2)). Recursion does not always need backtracking. Related question: Recursion vs. Now, we can consider countBinarySubstrings (), which calls isValid () n times. Any recursive solution can be implemented as an iterative solution with a stack. This is the recursive method. First, let’s write a recursive function:Reading time: 35 minutes | Coding time: 15 minutes. While tail-recursive calls are usually faster for list reductions—like the example we’ve seen before—body-recursive functions can be faster in some situations. Quoting from the linked post: Because you can build a Turing complete language using strictly iterative structures and a Turning complete language using only recursive structures, then the two are therefore equivalent. ) Every recursive algorithm can be converted into an iterative algorithm that simulates a stack on which recursive function calls are executed. With your iterative code, you're allocating one variable (O (1) space) plus a single stack frame for the call (O (1) space). Recursion can be hard to wrap your head around for a couple of reasons. The base cases only return the value one, so the total number of additions is fib (n)-1. For medium to large. While a recursive function might have some additional overhead versus a loop calling the same function, other than this the differences between the two approaches is relatively minor. How many nodes are. Now, one of your friend suggested a book that you don’t have. Recursion shines in scenarios where the problem is recursive, such as traversing a DOM tree or a file directory. Understand Iteration and Recursion Through a Simple Example In terms of time complexity and memory constraints, iteration is preferred over recursion. In this post, recursive is discussed. . Recursion adds clarity and (sometimes) reduces the time needed to write and debug code (but doesn't necessarily reduce space requirements or speed of execution). Time complexity: It has high time complexity. Time Complexity: O(n) Space Complexity: O(1) Note: Time & Space Complexity is given for this specific example. Comparing the above two approaches, time complexity of iterative approach is O(n) whereas that of recursive approach is O(2^n). 2. Recursion tree would look like. Time Complexity: O(N) Space Complexity: O(1) Explanation. So a filesystem is recursive: folders contain other folders which contain other folders, until finally at the bottom of the recursion are plain (non-folder) files. g. By the way, there are many other ways to find the n-th Fibonacci number, even better than Dynamic Programming with respect to time complexity also space complexity, I will also introduce to you one of those by using a formula and it just takes a constant time O (1) to find the value: F n = { [ (√5 + 1)/2] ^ n} / √5. If the structure is simple or has a clear pattern, recursion may be more elegant and expressive. This paper describes a powerful and systematic method, based on incrementalization, for transforming general recursion into iteration: identify an input increment, derive an incremental version under the input. If the algorithm consists of consecutive phases, the total time complexity is the largest time complexity of a single phase. By breaking down a. The Iteration method would be the prefer and faster approach to solving our problem because we are storing the first two of our Fibonacci numbers in two variables (previouspreviousNumber, previousNumber) and using "CurrentNumber" to store our Fibonacci number. The order in which the recursive factorial functions are calculated becomes: 1*2*3*4*5. With this article at OpenGenus, you must have a strong idea of Iteration Method to find Time Complexity of different algorithms. Looping will have a larger amount of code (as your above example. With regard to time complexity, recursive and iterative methods both will give you O(log n) time complexity, with regard to input size, provided you implement correct binary search logic. In graph theory, one of the main traversal algorithms is DFS (Depth First Search). To know this we need to know the pros and cons of both these ways. Binary sorts can be performed using iteration or using recursion. Loops are the most fundamental tool in programming, recursion is similar in nature, but much less understood. Note: To prevent integer overflow we use M=L+(H-L)/2, formula to calculate the middle element, instead M=(H+L)/2. Time and space complexity depends on lots of things like hardware, operating system, processors, etc. Second, you have to understand the difference between the base. (loop) //Iteration int FiboNR ( int n) { // array of. In simple terms, an iterative function is one that loops to repeat some part of the code, and a recursive function is one that calls itself again to repeat the code. 1) Partition process is the same in both recursive and iterative. Iteration produces repeated computation using for loops or while. Its time complexity is fairly easier to calculate by calculating the number of times the loop body gets executed. Space complexity of iterative vs recursive - Binary Search Tree. Obviously, the time and space complexity of both. Recursion is not intrinsically better or worse than loops - each has advantages and disadvantages, and those even depend on the programming language (and implementation). The speed of recursion is slow. So whenever the number of steps is limited to a small. High time complexity. what is the major advantage of implementing recursion over iteration ? Readability - don't neglect it. Iteration is a sequential, and at the same time is easier to debug. Proof: Suppose, a and b are two integers such that a >b then according to. hdante • 3 yr. Its time complexity is easier to calculate by calculating the number of times the loop body gets executed. There is less memory required in the case of. Apart from the Master Theorem, the Recursion Tree Method and the Iterative Method there is also the so called "Substitution Method". It may vary for another example. Transforming recursion into iteration eliminates the use of stack frames during program execution. . Recursion is more natural in a functional style, iteration is more natural in an imperative style. Both approaches create repeated patterns of computation. Both approaches create repeated patterns of computation. So let us discuss briefly on time complexity and the behavior of Recursive v/s Iterative functions. There’s no intrinsic difference on the functions aesthetics or amount of storage. Your understanding of how recursive code maps to a recurrence is flawed, and hence the recurrence you've written is "the cost of T(n) is n lots of T(n-1)", which clearly isn't the case in the recursion. Loops are almost always better for memory usage (but might make the code harder to. We have discussed iterative program to generate all subarrays. Additionally, I'm curious if there are any advantages to using recursion over an iterative approach in scenarios like this. Here we iterate n no. If it's true that recursion is always more costly than iteration, and that it can always be replaced with an iterative algorithm (in languages that allow it) - than I think that the two remaining reasons to use. Recursion: Analysis of recursive code is difficult most of the time due to the complex recurrence relations. Iteration Often what is. A recursive structure is formed by a procedure that calls itself to make a complete performance, which is an alternate way to repeat the process. There are O(N) iterations of the loop in our iterative approach, so its time complexity is also O(N). An iterative implementation requires, in the worst case, a number. This way of solving such equations is called Horner’s method. Because of this, factorial utilizing recursion has. In the recursive implementation on the right, the base case is n = 0, where we compute and return the result immediately: 0! is defined to be 1. In our recursive technique, each call consumes O(1) operations, and there are O(N) recursive calls overall.